Graph the function. \[ f(x)=\left\{\begin{array}{ll} |x| & \text { for } x<1 \\ -x+2 & \text { for } x \geq 1 \end{array}\right. \]
Solution
Step 1 ：First, we need to understand the function. It is a piecewise function with two parts. The first part is the absolute value function for \(x<1\), and the second part is a linear function for \(x \geq 1\).
Step 2 ：Let's graph the first part of the function, which is \(|x|\) for \(x<1\). The absolute value function is a V-shaped graph that intersects the origin. However, since it's only for \(x<1\), we only draw the left part of the V shape.
Step 3 ：Next, we graph the second part of the function, which is \(-x+2\) for \(x \geq 1\). This is a straight line with a slope of -1 and a y-intercept of 2. Since it's only for \(x \geq 1\), we start the line at the point where \(x=1\) and \(y=1\), and extend it to the right.
Step 4 ：Finally, we combine the two parts of the graph to get the graph of the entire function. The graph should look like a V shape on the left for \(x<1\) and a straight line sloping downwards for \(x \geq 1\).
